Chalkdust’s Category Theory

[Deep] connections that exist between other branches of mathematics … are mathematics, and that mathematics has a name: category theory.

What is a category?

[A] category is a collection of objects with relationships between them (called morphisms) that behave nicely in terms of composition and associativity. This provides a template for mathematics, and depending on what you feed into that template, you’ll recover one of the mathematical realms

“There is hardly any species of mathematical object that doesn’t fit into this convenient, and often enlightening, template.” Indeed, … “category theory is the mathematics of mathematics”.

It’s all about relationships

One of the main features of category theory is that it strips away a lot of detail: it’s not really concerned with the individual elements … .

[One] of the main maxims of category theory is that a mathematical object is completely determined by its relationships to all other objects. To put it another way, two objects are essentially indistinguishable if and only if they relate to every object in the category in the same way. [Cf Yoneda lemma ].

Since its inception, category theory has found natural applications in computer science, quantum physics, systems biology, chemistry, dynamical systems, and natural language processing, just to name a few. (The website `Applied category theory’ contains a list of applications.)

Conclusion

[Thinking] categorically can help serve as a beacon—it can strengthen your intuition and sharpen your insight—as you trek through the nooks and crannies of your favourite mathematical realms. And these days it’s especially hard to escape the pervasiveness of category theory throughout modern mathematics. So whatever your mathematical goals may be, learning a bit about categories will be well worth your time!

Comments

Interpretation

The Yoneda Lemma applies to ‘locally small’ categories, where the morphisms are ‘sets’, that is ‘a well-defined collection of distinct objects’. The website `Applied category theory’ notes:

These are all subjects in which I have or have had an interest. There seems to be at least a hint that mathematics and category theory, like set theory, is about ‘well-defined’ things, whereas the areas of application are not. Some people seem to find this confusing, and even claim that mathematics is (necessarily) inappropriate or at least misleading or dangerous when applied to these areas (if only because of then potential for confusion). Under ‘its all about relationships’ we are told that

[One] of the main maxims of category theory is that a mathematical object is completely determined by its relationships to all other objects. To put it another way, two objects are essentially indistinguishable if and only if they relate to every object in the category in the same way.

What should we make of this?

One reading is that category theory (and hence mathematical models and more broadly mathematics has nothing at all to say about the subject domains as such, but simply provides a logical framework in which domain-specific theories can be expressed. That is, category theory is at most a ‘theory about theories’ and not at all a theory about any external reality. This seems quite reasonable, but

In many domains, for example economics, the existence of a mathematical theory is somehow taken to imply some sort of validity in its application, which under this interpretation obviously isn’t true.

If – as we often seem to – we regard mathematical models as the hallmark of a good theory, then practitioners will need to develop a ‘well-defined’ theory, which (at least to me) seems an unreasonable constraint.

Another possible reading is that the article is actually about ‘the theory of locally small categories’. If we start from the view that the subject domains are not necessarily locally small then this begs some questions:

Where is the guidance on reducing an application-domain theory to such a locally small theory?

How can we be sure that such a reduction doesn’t lose something in the process?

Don’t we, at least in principle, need to start from a larger category theory, maybe quasi-categories?

Depending on the reading, it may be that (as many suppose):

Mathematics provides only a very limited view of the application domains. For example, many of the application domains are inherently unstable and hence not ‘well-defined’: they only appear temporarily stable because they are made so by some engineering process, which would seem to be outside the purview of mathematics. Thus mathematics is very limited and misleading in what it can say about real-world phenomena: it can only formalise what practitioners think.

Small worlds

The reference to ‘locally small’ versus ‘large’ categories resonates with Savage’s distinction between small and large worlds. Perhaps:

Mathematics is only fits for small worlds.

In this case, many of the issues I have come across in the application of mathematics (and science) might be explained by supposing that practitioners are having to treat their large worlds as small in order to make use of the ‘accepted’ methods, and that sometimes this is dangerously misleading. To extrapolate from Lindley’s view:

[It is wrong to apply conventional ‘scientific’ theories methods to large worlds as if they are small.} It is essential to use a small world, which introduces simplification but often causes distortion. … Where a real difficulty arises is in the construction of the model … there is a real gap in our appreciation of how to assess probabilities-of how to express our uncertainties in the requisite form. My view is that the most important … research topic …. is the development of sensible methods. This will require co-operation with numerate [practitioners] and much experimental work.

Thus mathematical models – in this reading – can only be small-world models, and hence must be silent on any large-world issues.

Radical uncertainty

An alternative view (as in my blog) is to take a wider view of mathematics, considering categories that are not necessarily ‘well-defined’ but simply constrained. These need to allow for what has been called ‘radical uncertainty’. Such a category theory might then have something to say about the application domains as large worlds, rather than being confined to (inadequate and misleading) small-world theories of large worlds.